Multiple lp-mode fiber designs for mode-division multiplexing

ABSTRACT

A few-mode fiber is described, having a graded-index core and a surrounding cladding comprising a ledge between the core and the trench, a down-doped trench abutting the ledge, and an undoped cladding region abutting the trench. The fiber&#39;s refractive index profile is configured to support 9 or more LP modes for transmission of a spatially-multiplexed optical signal. Undesired modes have respective effective indices that are close to, or less than, the cladding index so as to result in leakage of the undesired modes into the outer cladding. The index spacing between the desired mode having the lowest effective index and the leaky mode with the highest effective index is sufficiently large so as to substantially prevent coupling therebetween.

CROSS REFERENCE TO RELATED APPLICATIONS

The present application claims the priority benefit of U.S. Prov. Pat.App. No. 62/060,138, filed on Oct. 6, 2014.

The present application is a continuation-in-part of U.S. patentapplication Ser. No. 13/838,981, which was filed on Mar. 15, 2013, andwhich was published as United States Pat. Pub. No. 2014/0064686 on Mar.6, 2014.

U.S. patent application Ser. No. 13/838,981 claims priority from U.S.Prov. App. No. 61/696932 filed Sep. 5, 2012.

The above applications are owned by the assignee of the presentinvention and are incorporated herein by reference in their entirety.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The invention relates to optical fibers designed for space-divisionmultiplexing (SDM). More specifically it relates to optical fibers thatefficiently transmit optical signals in multiple modes withoutsubstantial crosstalk.

2. Background of the Invention

(The following may or may not constitute prior art.)

Previous work on multiple mode optical fibers for mode-divisionmultiplexing focused on step and graded index (GRIN) fiber designs foroptimized two LP mode fibers (having three spatial modes). We have alsodisclosed GRIN fiber designs with 1% core relative deltas, including ashelf and trench in the cladding, which are optimized to support four LPmodes (having six spatial modes).

SUMMARY OF INVENTION

We have now designed GRIN fibers with lower core relative delta (near0.8%) which have desirable properties for transmission. These lowerdelta fibers will have lower attenuation losses due to reduced Rayleighscattering, which is desirable to improve performance. We have alsodesigned fibers with optimized-raised triangle, depressed-claddingprofiles to support two and four LP modes. Recently work on fibersdesigned to support space-division multiplexing (SDM) has been reported.This work has generally focused either on fibers that contain multiplecores with weak coupling between the cores or on fibers with a singlecore that supports the propagation of a few modes.

An aspect of the invention is directed to a few-mode fiber having agraded-index core and a surrounding cladding. The cladding comprises aplurality of regions, including a ledge, a down-doped trench abuttingthe ledge, and an undoped outer cladding abutting the trench. Thefiber's refractive index profile is configured to support 9 or more LPmodes for transmission of a spatially-multiplexed optical signal.Undesired modes have respective effective indices that are close to, orless than, the cladding index so as to result in leakage of theundesired modes into the outer cladding. The index spacing between thedesired mode having the lowest effective index and the leaky mode withthe highest effective index is sufficiently large so as to substantiallyprevent coupling therebetween.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is an illustrative diagram of modal propagation constants for ahypothetical step-index or graded-index few-mode fiber (FMF) with nocladding structure;

FIG. 2 illustrates the index of refraction for step-index (α=∞) andparabolic (α=2) core shapes;

FIG. 3 shows the Normalized Propagation Constant, b_(l,m), as a functionof V for LP_(l,m) modes of a step-index fiber. Vertical red line islocated at V=3.15 where the differential group delay (DGD) between theLP₁₁ and LP₀₂ modes is zero;

FIG. 4 shows group delay as a function of V for the first 6 LP_(l,m)modes of a step-index fiber;

FIG. 5 shows normalized propagation constants for the first 6 modes of aparabolic core;

FIG. 6 shows group delay curves for 6 modes of a parabolic (α=2.00) corefiber;

FIG. 7 shows a four-quadrant chart illustrating the strength of modecoupling in a deployed FMF transmission line;

FIG. 8 illustrates a raised-triangle, depressed-cladding index profilethat is optimized for low DGD and ease of manufacturing;

FIG. 9 shows the calculated differential group delay between the LP₀₁and LP₁₁ modes over a wide range of V-numbers for a prototype fiberhaving a raised-triangle, depressed-cladding design;

FIG. 10 shows the normalized propagation constant as a function ofV-number of a raised-triangle, depressed cladding design;

FIG. 11 illustrates a raised triangle profile based on actual VAD coreshape;

FIG. 12 shows a refractive index profile for a practice of theinvention, Example 2.

FIG. 13 shows the differential group delay for the optical fiber ofExample 2;

FIG. 14 shows the normalized propagation constant versus V-number fortwo families of fiber profiles;

FIG. 15 shows group delays of LP₀₁, LP₁₁, LP⁰², LP₂₁, LP⁰³ and LP₁₂modes for a parabolic GRIN-FMF (lower curves) and a GRIN-FMF with acladding structure, optimized for operation in the C-band (uppercurves);

FIG. 16 is a refractive index profile for another practice, Example 3,of the invention;

FIG. 17 gives differential group delay data for Example 3;

FIG. 18 is a refractive index profile for another practice, Example 4,of the invention;

FIG. 19 gives differential group delay data for Example 4;

FIG. 20 is a refractive index profile for another practice, Example 5,of the invention;

FIG. 21 gives differential group delay data for Example 5;

FIG. 22 is a refractive index profile for another practice, Example 6,of the invention, which is a six-mode design;

FIG. 23 gives differential group delay between the first fivehigher-order LP modes and the LP₀₁ mode for Example 6;

FIG. 24 shows a refractive index profile for the design of Example 7, anUltra Large Area few-mode fiber (ULA-FMF) design;

FIG. 25 shows differential group delay between the LP₁₁ mode and theLP₀₁ mode for Example 7;

FIG. 26 shows a refractive index profile for the design of Example 8, asecond Ultra Large Area few-mode fiber (ULA-FMF) design;

FIG. 27 shows differential group delay between LP₁₁ mode and LP₀₁ modefor Example 8;

FIG. 28 shows a refractive index profile for the design of Example 9, athird Ultra Large Area few-mode fiber (ULA-FMF) design;

FIG. 29 shows differential group delay between LP₁₁ mode and LP₀₁ modefor Example 9;

FIGS. 30 and 31 show, respectively, cross section and isometric views ofan exemplary FMF according to a further aspect of the invention (Example10);

FIG. 32 shows a table setting forth design parameters for Regions 1-4 ofExample 10;

FIG. 33 shows a refractive index profile for Example 10;

FIG. 34 shows a graph illustrating, for each of the higher-order modessupported by Example 10, the mode's predicted index difference relativeto an undoped silica cladding, and the mode's predicted differentialgroup delay;

FIG. 35 shows a table illustrating the theoretical properties of Example10;

FIG. 36 shows a graph illustrating the target refractive index profileand the actual refractive index profile achieved in a first attempt tofabricate a prototype of Example 10;

FIG. 37 shows a graph illustrating the target refractive index profileof the graded-index core region, and the measured refractive indexprofile for Example 10;

FIG. 38 shows a graph illustrating the relationship between the preformindex for the graded-index core of Example 10 and the germania flow rate(expressed in arbitrary units);

FIG. 39 shows a graph of the output of an S2 measurement on a 20-metersample of Example 10, and FIG. 40 shows a graph of the results;

FIG. 41 shows a table setting forth the measured modal loss for thefour-mode groups of Example 10;

FIGS. 42-45 set forth data for a further example of a 9 LP mode FMFaccording to the invention (Example 11-1);

FIGS. 46-49 set forth data for another example of a 9 LP mode FMFaccording to the invention (Example 11-2);

FIGS. 50-53 set forth data for a further example of a 9 LP mode FMFaccording to the invention (Example 11-3);

FIGS. 54 and 55 set forth data for another example of a 12 LP mode FMFaccording to the invention (Example 12);

FIGS. 56 and 57 set forth data for another example of a 16 LP mode FMFaccording to the invention (Example 13);

FIGS. 58 and 59 set forth data for another example of a 20 LP mode FMFaccording to the invention (Example 14); and

FIGS. 60A-60F are a series of figures illustrating the sensitivity ofDGD between the LP₀₁ and LP₁₁ modes to variations in a number of designparameters for an exemplary 2 LP mode FMF.

DETAILED DESCRIPTION

Interest in SDM is mainly due to the impending “capacity crunch,” inwhich the fundamental, non-linear Shannon limit to increasing thespectral efficiency of fiber optic transmission will force carriers todeploy fiber cables at an accelerating rate, rather than simplydeploying faster transmitters at decreasing marginal cost-per bit, thusdestroying the economics of the backbone network. A rich new medium with100 to 1000 times the capacity of standard single mode fiber (SSMF)would be required. Few-mode fiber (FMF) technology combined withmulti-core fiber technology might create such a medium.

For use in high-capacity SDM transmission it is desirable that thewaveguide:

-   -   supports the low loss propagation of N unique modes, where N is        at least 2 and possibly 10 to 20. Here, “low loss” is considered        to be that of conventional single mode fiber.    -   has low differential mode attenuation (DMA), for example less        than about 0.02 dB/km. DMA is a fundamental, uncorrectable        impairment that limits the capacity of transmission based on        multiple-input, multiple output (MIMO) signal processing.    -   provides low differential group delay (DGD) between all of the        low loss modes so that the receiver design can be simplified. To        support 1000 km transmission with ASIC technology that may be        realizable on a 10-year timeframe, the accumulated DGD of a FMF        transmission line probably needs to be equalizable with perhaps        hundreds of T/2-spaced complex taps for a time domain equalizer.        This represents a technological, but not a fundamental,        limitation. The relationship between fiber DGD and accumulated        DGD will be discussed below. In a contrary view, large DGD may        have the beneficial impact of reducing non-linear crosstalk        between modes. (As used herein, the term “differential group        delay” or “DGD” as it relates to LP modes carried by an FMF is        generally synonymous with the term “differential mode delay” or        “DMD.”)    -   optimizes the strength of distributed mode coupling. It has been        proposed that low mode-coupling in the fiber will minimize the        complexity of MIMO crosstalk mitigation hardware. In a contrary        view, strong mode coupling has the benefit of minimizing the        accumulation of DGD with distance as well as minimizing the        impact of DMA in the fiber and mode-dependent gain in the        amplifier.    -   provides a low level of transmission penalty caused by nonlinear        propagation impairments, including maximizing the effective        areas of the low loss modes.    -   can be cost effectively realized with state-of-the-art fiber        fabrication techniques. It will be noted that alternative        suggestions have been put forward as to the most beneficial        properties for FMFs. In the following portion of the        specification we will discuss FMF design strategies for        step-index and graded index fibers and consider the inevitable        tradeoffs that will be made in trying to achieve a design that        meets any set of objectives. Recent fiber design and        transmission experiments over few-mode fiber have been conducted        with two limits in mind. In one case, it is assumed that mode        coupling in an N-mode fiber will be confined to a subset of M        modes where M<N. In one example of this low mode-coupling        paradigm, the fiber supported five spatial modes (comprising        LP₀₁, LP₁₁, and LP₂₁), where the only strong couplings were        between LP₁₁a and LP₁₁b and then between LP₂₁a and LP₂₁b. So it        was only necessary to implement two 4×4 MIMO recovery algorithms        instead of one 10×10 MIMO algorithm. In another paradigmatic        case, it is assumed that all N fiber spatial modes mix        sufficiently such that full 2N×2N MIMO recovery of the signals        is necessary (where 2N accounts for two polarizations for each        spatial mode). In prior work independent data streams were        multiplexed onto three independent spatial modes (comprising        LP₀₁, LP₁₁) and their x- and y-polarizations were then        demultiplexed by a 6×6 MIMO recovery algorithm over 10 km, then        96 km, and then 1200 km. In the case of 96 km transmission,        large penalties were observed for reduced complexity 4×4 MIMO        demodulation of the LP_(11a) and LP_(11b) modes. This is the        more complex case for receiver design and implementation. In the        former case, the fiber design must maintain a relatively large        Δβ between nearest neighbor modes to reduce distributed mode        coupling. In the latter case, a smaller Δβ between guided modes        would beneficially slow the accumulation of DGD as a function of        length and mitigate the effects of DMA through stronger mode        mixing (within and between the low-loss modes).

In this context, it is interesting as well as important to consider thatDMA, mode coupling, and maximum accumulated DGD are inter-relatedquantities. Consider the schematic diagram of fiber mode propagationconstants in FIG. 1. (The propagation constant can be converted toeffective index by n_(eff)=β/k.) Modes above the cladding index arebound modes, while those below the cladding are leaky modes. Attenuationand DMA will be strongly influenced by the macro- and microbendinglosses of the modes. Macrobending loss is generally minimized bymaximizing (or equivalently the effective index n_(eff)); keepingβ_(min) above some minimum level is critical for minimizing loss in SSMFand DMA in FMF. SSMF is generally designed so that there is a highlylossy, though technically still bound, mode just above the claddingindex; that mode is said to be effectively cutoff through the high loss.The same principle will hold for step-index or graded-index FMF with nocladding structure: an optimized design would have a mode just above thecladding index as shown as a dashed line in FIG. 1. In the case of astructured cladding, the lossy mode may drop below the cladding indexand become a leaky mode. It is difficult in practice to engineer themodal spacings such that Δβ_(lossy)>>Δβ_(i,j).

In the presence of cable stress, microbending loss may result fromcoupling between bound modes (i.e., the modes that carry data signals)and the lossy or leaky modes. Microbending loss for the lowest boundmode is a strong function of Δβ_(lossy), so maximizing Δβ_(lossy) isalso a condition for minimizing DMA. In an ideal case for the strongmode-mixing paradigm, all Δβ_(i,j) would be small, β_(min) would berelatively large, and Δβ_(lossy)>>Δβ_(i,j). This would result in a fiberwhere modes couple strongly in pairwise fashion leading to (1)accumulation of DGD which is proportional to the square root of thefiber length even over shorter links of a few hundred km and (2)mitigation of the deleterious impact of DMA. Furthermore macro- andmicrobending of the lowest guide mode would be small, leading to lowDMA. In fact, these are difficult conditions to fulfill. It is typicalthat the spacing between adjacent modes does not vary strongly over afew modes, and there is typically no abrupt change in mode spacingacross the cladding index. In other words, it is challenging to design afiber to promote mixing between multiple low-loss bound modes whilesimultaneously minimizing the loss of the lowest bound mode.

Since low DMA is a fundamental requirement, we conclude that β_(min)must be kept greater than some threshold for low macrobending andΔβ_(lossy) (typically similar to Δβ_(i,j)) must be kept large enough tominimize microbending loss. Once these two criteria are fulfilled, therewill typically be little flexibility to manipulate the magnitude ofΔβ_(i,j).

Consider a circularly symmetric optical fiber with cladding of infiniteradial extent and radially varying index of refraction as shown in FIG.2. The index of the cladding is given by n_(clad) and the index of thecore at r=0 is n_(core). The index within the core, n(r), at radialposition r is given by

$\begin{matrix}{{n(r)} = {{n_{clad} + {n_{core}*\left\lbrack {1 - \left( \frac{r}{a} \right)^{\propto}} \right\rbrack \mspace{14mu} {for}\mspace{14mu} r}} \leq a}} & (1)\end{matrix}$

where a is the core radius, α is the core shape parameter. The idealstep-index core shape occurs when α becomes infinite.

It can be shown that the effective index, β/k, of a mode guided by thiswaveguide structure must satisfy the inequality

n _(clad) <β/k<n _(core)   (2)

where β is the propagation constant of the mode and k=2π/λ is thepropagation constant of a plane wave in free space. When the effectiveindex is greater than the cladding index the solutions for thetransverse fields in the cladding region are radially evanescent andtherefore the modal energy is confined within the waveguide structureand the mode is referred to as a guided mode. A mode is said to becutoff when its effective index is equal to the cladding index since thesolutions for the transverse fields in the cladding is oscillatory,rather than evanescent, and energy is carried away from the fiber axis.In general it is desirable for a mode to have effective index far abovethe cladding index since this results in rapid decay of the evanescentfield in the cladding, and it being less susceptible to bending losses.

When the weakly guiding assumption holds, i.e., when

${\frac{n_{core} - n_{clad}}{n_{clad}}1},$

then the waveguide properties can be accurately approximated by linearlypolarized modes that have no longitudinal field components, i.e., thepolarization is in the plane transverse to the fiber axis. The fieldsand characteristic equation of the linearly polarized modes can bedescribed by simple analytic formulas that simplify calculation of thewaveguide properties. The properties of the LP modes are a goodapproximation of those of the real modes of weakly guiding fibers over awide range of conditions. For these reasons, the LP mode analysis isoften used when considering typical optical fibers used in opticalcommunications systems.

The LP modes correspond to degenerate groups of the HE, TE and TM modesgiven by the more general analysis that does not make use the weaklyguiding approximation. For the LP modes with no azimuthal variation ofthe fields, i.e., the azimuthal mode number is zero, the LP modes arecomprised of two degenerate modes; the two polarizations of the HE_(1x)modes. For the LP modes with azimuthal variation of the fields, i.e.,the azimuthal mode number is greater than zero, then the LP modes arecomprised of four nearly degenerate modes; a set of HE, EH, TE and TMmodes.

The LP_(l,m) nomenclature is generally used to name the individuallinearly polarized modes. Here, the azimuthal and radial mode-numbersare given by l and m, respectively. The lowest order LP₀₁ mode is oftenreferred to as the “fundamental mode” and corresponds to the twopolarizations of the HE₁₁ mode. The first higher-order mode, the LP₁₁mode, is comprised of the two polarizations of the HE₂₁ mode and theTM₀₁ and TE₀₁ modes, i.e., four nearly degenerate “real” modes.

The normalized frequency of a step-index fiber is defined as

V=k a(n _(core) ² −n _(clad) ²)^(1/2) ≈k n _(core) a√{square root over(2Δ)}  (3)

where

$\Delta = {\frac{n_{core} - n_{clad}}{n_{clad}}.}$

The normalized frequency is sometimes referred to as the waveguidestrength because any given guided mode will be better confined to thecore, i.e., more strongly guided, when the waveguide has a larger valueof V.

FIG. 3 shows the normalized propagation constant of the guided LP_(l,m)modes of a step-index fiber as a function of the normalized frequency,V. The normalized propagation constant of the i,j mode, b_(i,j), isdefined as

$\begin{matrix}{b_{i,j} = \frac{\left( {\frac{\beta_{i,j}}{k} - n_{clad}} \right)}{\left( {n_{core} - n_{clad}} \right)}} & (4)\end{matrix}$

when V is less than 2.405, then only the fundamental LP₀₁ satisfies thecondition that the effective index is greater than n_(clad) andtherefore the fiber is single-moded. When V is greater than 2.405 thenadditional modes satisfy the propagation condition and the fibersupports the propagation of more than one LP mode.

When designing the index profile of a single mode fiber it is usual toplace the V value slightly greater than 2.405 at the shortest operatingwavelength, say V˜2.8. Even though the fiber can theoretically supportthe propagation of the LP₁₁ mode, the effective index of the LP₁₁ modeis very low and the loosely bound LP₁₁ mode is susceptible to excessloss caused by bending and waveguide imperfections. With a fiber of thisdesign under practical deployment conditions, the LP₁₁ mode iseffectively cut off because of the excess losses that result frombending. This design trick of operating the waveguide at V-numberslightly above the cutoff V-number results in a “stronger waveguide” andtherefore the fundamental mode has better mode confinement and lowersusceptibility to bending loss than would be otherwise possible. Thissame design approach can be used when designing FMFs.

As noted previously, it is desirable for FMFs to have low mode couplingbetween the modes that will be used for SDM multiplexing to minimize thecrosstalk between the multiplexed data streams. An additionalrequirement is that the highest-order mode used in the SDM scheme shouldhave low mode coupling to guided, leaky, or radiation modes of an evenhigher order since energy coupled to these modes results in energy loss.

The field shapes of the guided modes of an ideal fiber satisfy anorthogonality condition and therefore energy does not couple between themodes. However, in a real fiber the orthogonality can be broken byimperfections in the fiber, e.g. inhomogeneities of the index ofrefraction or deformations of the fiber axis or core size, corenoncircularity, etc.; which can result in the coupling of energy betweenthe modes. Imperfections in the transmission path or coupling points cancause optical modes to exchange power. This issue can be addressed withMIMO signal processing, but for a good understanding of the FMFproperties, one must have a grasp of the potential and implications ofmode coupling.

For degenerate modes (such as the two polarizations of the LP₀₁ whichhave identical phase constants) the mode coupling is usually strong;that is a substantial optical power will be transferred between themodes within a few tens of meters. In the case of other modes (LP₁₁ toLP₀₁ for example) the coupling can be much weaker, and depends on therelative difference in phase constants. In such a case, the opticalsignal may travel tens of kilometers before there is significantcoupling to another mode. Different FMF design strategies can result ineither strong or weak mode coupling. Prior work found that energy willcouple between two modes when the imperfections have a longitudinalspatial frequency component equal to the difference in the longitudinalpropagation constants of the modes, Δβ. The strength of the couplingbetween two modes is a strong function of Δβ. Coupling between modes ofadjacent mode groups is proportional to

(Δβ)^(−(4+2p))   (5)

where p characterizes the power spectrum of the perturbation andtypically has values of 0, 1 or 2 depending on the nature of theexternal stresses, the fiber outer diameter, and coating properties.This result implies that to minimize mode coupling we must maximize theΔβ of the modes.

From FIG. 3 one can see for a step-index 2-mode fiber with V equal toabout 4 that Δβ between the LP₀₁ and LP₁₁ modes and Δβ between the LP₁₁and the LP₂₁ modes are simultaneously maximized. This condition resultsin low mode coupling between the LP₀₁ and LP₁₁ modes that are used forSDM and low mode coupling between the LP₁₁ and the lossy LP₂₁ mode.Similarly, for a 4-mode step-index fiber that supports propagation ofthe LP₀₁, LP₁₁, LP₂₁ and LP₀₂ modes, the mode coupling between the 4modes will be minimum when the V˜5.5. However, the coupling between LP₂₁and LP₀₂ will always be relatively much stronger than coupling betweenother pairs of modes.

When the group velocities of the modes that carry independent SDM datachannels are different, then pulses that are simultaneously launchedinto the various modes of the fiber will arrive at the end of the fiberat different times. When mode coupling and DGD are both present thencrosstalk between modes can spread across multiple bit periods. The MIMOsignal processing electronics that address channel crosstalk in the SDMreceiver hardware become more complex when the accumulated DGD betweenthe modes grows and the crosstalk spreads over many bit periods.Therefore for long distance SDM transmission it is desirable to minimizethe DGD.

FIG. 4 plots the normalized group delay as a function of V for variousmodes of a step-index fiber. FIG. 4 shows that the group delay curves ofthe LP₀₁ and the LP₁₁ modes cross and the DGD becomes zero when V isapproximately equal to 3.15. Note that we found in the previous sectionthat the Δβ's of a 2-mode step-index fiber are maximized when V˜4. Sofor 2-mode step-index fibers it is not possible to simultaneouslyminimize both DGD and mode coupling. For a 4-mode fiber, FIG. 3 showsthat a step-index design does not exist where the group delay betweenall of the lowest order 4-modes is zero. Note that for step-index fibersthere are values of V where the group delays of a subset of the guidedmodes are equalized. For example the group delay of the LP₀₂, LP₂₁ andLP₁₂ modes are approximately equal when V is equal to about 6.5.However, when V˜6.5, the fiber supports three more modes with quitedifferent group delays.

In FIG. 4 the vertical line is located at V=3.15 where the DGD betweenthe LP₀₁ and LP₁₁ modes is zero. However, when V=3.15 the normalizedpropagation constant of the LP₁₁ mode is very small and coupling betweenthe LP₁₁ mode and leaky modes will be large resulting in DMA. WhenV˜4.5, the normalized propagation constants and the difference betweenthe propagation constants of the LP₀₁ and LP₁₁ are large which gives lowsensitivity to mode coupling between the LP₀₁ and LP₁₁ modes and betweenthe LP₁₁ mode and leaky modes. Further, the propagation constant of theLP₀₂ and LP₂₁ modes is very small so that these modes will be very lossyand therefore only the two lowest order modes propagate with low loss.However, the LP₀₁ and LP₁₁ mode DGD is quite large when V˜4.5. Inaddition to the magnitude of the Δβ of the fiber profile design, modecoupling also depends on factors related to the deployment of the fiber.Here cabling and splicing effects need to be considered. Cabling stresswill increase distributed mode coupling by providing an additionalsource of perturbations of the fiber. Splices and connectors providepoints of discrete mode coupling.

When small and random mode coupling is considered, it can be shown thatthe DGD will grow linearly with length for distances much shorter thanthe correlation length and as the square root of length for longlengths. The two-mode case is completely analogous to the resultsobtained for PMD. If a short pulse is launched simultaneously in eachmode then the variance in arrival times of portions of the pulse isgiven as a function of fiber length, L:

$\begin{matrix}{{\langle\left( {T - {\langle T\rangle}_{av}} \right)^{2}\rangle}_{av} = {\frac{{DGD}^{2}l_{c}L}{4}\left\lbrack {1 - {\frac{l_{c}}{2L}\left( {1 - {\exp \left( {{- 2}{L/l_{c}}} \right)}} \right)}} \right\rbrack}} & (6) \\{{\lim_{{L/l_{c}}->\infty}{\langle\left( {T - {\langle T\rangle}_{av}} \right)^{2}\rangle}_{av}} = \frac{{DGD}^{2}l_{c}L}{4}} & (7)\end{matrix}$

where l_(c) is the correlation length and T is the time-of-flightthrough the fiber. Note from the second equation (long fiber limit) thatthe spread in arrival times scales as the square root of the product ofthe correlation length and the fiber length. A similar scaling law holdsfor guides with any number of modes.

The inability of two-mode, step-index fibers to simultaneously providelow DGD, low mode coupling and low DMA leads to consideration of fiberswith more complicated core shape. It was pointed out in prior work thatwhen the core shape parameter α is 2.5 that the group delay curves ofthe LP₀₁ and LP₁₁ modes cross when V is ˜5.5 and that the fiber iseffectively two-moded. FIG. 5 and FIG. 6 show curves of the normalizedpropagation constant and group delay, respectively, for the first 6 LPmodes of a parabolic (α=2.00) core fibers a function V. Fornon-step-index fibers, i.e. α≠∞, we define V as previously defined forstep-index fibers. When α is finite, the fiber is single moded whenV<2.405·(1+2/α)². For example, cutoff occurs when V=3.40 for a parabolicprofile. The core shape parameter α can be chosen to minimize DGD at aparticular wavelength.

FIG. 5 illustrates that when V˜6, the difference between the propagationconstants of the LP₀₁ and LP₁₁ are large which gives low sensitivity tomode coupling between the LP₀₁ and LP₁₁ modes. Also when V˜6 thenormalized propagation constant of the LP₁₁ mode of the parabolic corefiber is quite large which minimizes the coupling of LP₁₁ to leakymodes. Further, the propagation constant of the LP₀₂ and LP₂₁ modes arevery small so that these modes will be very lossy and therefore only thetwo lowest order modes propagate with low loss. FIG. 6 shows for aparabolic core shape that when V˜6 the difference between the LP₀₁ andLP₁₁ group delays is low.

FIG. 5 and FIG. 6 also show that when the V of a parabolic core shapefiber has value slightly larger than 6, the first four LP modes willhave widely spaced propagation constants giving low mode coupling aswell as low DGD. Further, the propagation constants of the LP₀₂ and LP₂₁modes are maximized while higher order modes are effectively cut-off.

When V˜6, the normalized propagation constants and the differencebetween the propagation constants of the LP₀₁ and LP₁₁ are large whichgives low sensitivity to mode coupling between the LP₀₁ and LP₁₁ modesand between the LP₁₁ mode and leaky modes. Further, the propagationconstant of the LP₀₂ and LP₂₁ modes are very small so that these modeswill be very lossy and therefore only the two lowest order modespropagate with low loss.

Also, when V˜6 and the difference between the LP₀₁ and LP₁₁ group delaysis low. (FIG. 6 shows group delay curves for 6 modes of a parabolic(α=2.00) core fiber. The difference between the group delay curves forthe LP₀₁ and LP₁₁ modes is small for V˜6.)

While two regimes of strong and weak mode coupling for few-modetransmission have been contemplated, there may be some doubt that a weakcoupling regime will exist in a deployed transmission link. The picturecan be clarified by considering that the strength of mode coupling in aFMF transmission line will depend on both distributed and discretecontributions. FIG. 7 shows a four-quadrant chart illustrating thepossibilities. In a deployed fiber cable splices will occurapproximately every five kilometers, so an 80 km amplified span willcontain about 16 splices on average. Furthermore, other components suchas wavelength selective switches and optical amplifiers will also benodes for mode-coupling.

If the mode coupling at splices is sufficient such that the correlationlength l_(c) is equal to five to 10 cable segments, then transmissionwill occur in a strongly mode-coupled regime regardless of the strengthof distributed mode coupling in the fiber (i.e. regardless of Δβ_(i,j)).This will have the beneficial result that DGD will accumulate as√{square root over (L)} in the link, and mitigate the impact of DMA, butnecessitate full 2N×2N MIMO processing in all cases. However large Δβwill nonetheless generally give the lowest possible DMA and perhapsalways be desirable for this fundamental reason. With reference to FIG.7, the strength of mode coupling in a deployed FMF transmission linewill depend on both distributed and discrete mode coupling. Transmissionin the weakly mode-coupled regime requires that both contributions beweak. If discrete mode coupling at splices, connections, and in-linecomponents is sufficiently strong, then the mode spacings in the fiberΔβ will be of secondary importance for mode-mixing considerations butremain of primary importance for minimizing DMA.

The impact of splicing modern FMF on mode coupling has not yet beenquantitatively determined, although early studies considered loss andmode-mixing at splices of traditional MMF. If it be the case thatsplices, connectors, and components generally leads to the strong modecoupling regime, then the upper right quadrant of FIG. 7 may prove to bethe best approach to FMF design, yielding lowest possible DGD and DMA,though necessitating full 2N×2N MIMO processing at the receiver.

Table I shows the modal content of the LP modes in terms of the morefundamental HE, TE, and TM modes. To calculate the total number of modesonto which data can be multiplexed, multiply by two to account for thetwo polarizations for each spatial mode pattern.

LP-Mode True Mode Number of Degenerate Designation Content Spatial ModesLP₀₁ HE₁₁ 1 LP₁₁ TE₀₁, TM₀₁, HE₂₁ 2 LP₂₁ EH₁₁, HE₃₁ 2 LP₀₂ HE₁₂ 1 LP₃₁EH₂₁, HE₄₁ 2 LP₁₂ TE₀₂, TM₀₂, HE₂₂ 2 LP₄₁ EH₃₁, HE₅₁ 2 LP₂₂ EH₁₂, HE₃₂ 2

The table illustrates that designing a FMF to support, e.g., 10 lowloss, orthogonal spatial modes is equivalent to designing for the lowest6 LP modes for transmission. Increasing the number of low loss modesrequires increasing the V-number. If V is increased by raising the corediameter, then the modes will become more closely spaced, the modeA_(eff) will increase, and Δβ_(lossy) will become smaller leading tohigher DMA. If V is increased by the increasing the core Δ, then themode A_(eff) will decrease, Rayleigh scattering losses will increase dueto higher concentration of GeO₂, and the modal spacing will increasehelping to minimize DMA. A judicious combination of adjusting core A anddiameter, along with other degrees of freedom in the profile, will benecessary to guide 10 to 20 modes with low DMA and low attenuationlosses.

FIG. 4 shows for a step-index fiber that the DGD between the LP₀₁ andLP₁₁ modes can be small only when the V-number is near 3.15, i.e. wherethe group delay curves cross. Therefore, to obtain low DGD with astep-index profile requires tight tolerances on core delta and coreradius to ensure that the V-number is close to 3.15. FIG. 4 also showsthat the group delay will remain low only over a narrow range ofwavelengths.

FIG. 8 illustrates a raised-triangle, depressed cladding index profilethat is optimized for low DGD and ease of manufacturing of two-modefiber over broad wavelength range and four-mode fiber over narrowwavelength range. In FIG. 8 n_(clad) is undoped silica (zero delta). Thecore comprises a portion extending from the center of the core,n_(core), to radius a, in which the refractive index decreases linearlyto point a. The maximum refractive index value of point a is greaterthan half of the value at n_(core). Abutting or adjacent the core is adown doped trench as shown. In FIG. 8, the trench is shown in contactwith the core. However, in some cases there may be a ledge between thecore and the trench.

The term “ledge” is used herein to define a region separating anup-doped core and a down-doped trench. Typically, the ledge portion isundoped.

In general terms the optical fiber just described can be characterizedas having a core and a cladding surrounding the core, wherein the coreand cladding have a refractive index profile that is structured tosupport propagation of a plurality of desired signal-carrying modes,while suppressing undesired modes, wherein the core comprises a portionextending from the center of the core, n_(core), to radius a, in whichthe refractive index decreases linearly from n_(core) to point a,wherein the cladding comprises a down-doped cladding region abutting oradjacent to the core, and an undoped cladding region abutting thedown-doped cladding region, wherein the core, and cladding areconfigured to support propagation of a spatially multiplexed opticalsignal comprising a plurality of desired modes, while suppressingundesired modes, wherein the core and surrounding cladding is configuredsuch that undesired modes have respective effective indices that areclose to or less than the cladding index so as to result in leaky modesthat leak into the outer cladding region, and wherein the index spacingbetween the desired mode having the lowest effective index and the leakymode with the highest effective index is sufficiently large so as tosubstantially prevent coupling therebetween.

FIG. 8 shows an index profile as in FIG. 8, referred to as araised-triangle, depressed-cladding design, that is optimized to provide2-mode operation with low DGD over a wider range of V than a step-indexprofile. The DGD of a two-mode fiber design with this profile shape isinsensitive to V-number.

FIG. 9 shows the group delay curves calculated for a prototyperaised-triangle, depressed-cladding fiber fabricated using the VADprocess. The group delay curves for the LP₀₁ and LP₁₁ modes fall veryclose to one another over a broad range of V-numbers. This behaviormaintains low DGD, i.e. less than 100 ps/km over the entire C-Band,while using standard fabrication techniques used for manufacturingsingle mode transmission fibers and manufacturing tolerances typical forSSMF. The VAD and rod-in-tube manufacturing techniques were used tofabricate a few hundred kilometers of raised-triangle,depressed-cladding, two-mode optical fiber with low DGD, low DMA andgood axial uniformity. The A_(eff) of the LP₀₁ and LP₁₁ modes were 155μm² and 160 μm², respectively. The attenuation of the two mode fiber was0.2 dB/km.

FIG. 10 shows normalized propagation constant as a function of V-numberof a raised-triangle, depressed cladding design. When V˜4.5, thewaveguide supports the propagation of two modes with low DGD asindicated in FIG. 9. The previous example shows that zero DGD can beachieved for two-mode fibers with step-index, parabolic index and raisedtriangle core depressed cladding index shapes. FIG. 4 shows that for astep-index fiber the group delay curves of all propagating modes crossonly for the two-mode case when V=3.15. FIG. 10 shows that, forparabolic core fiber, V can be chosen so that the group delay curves ofall but the highest order propagating mode lie very close together andtherefore low DGD is achievable. V is properly chosen when the highestorder mode is effectively cut off. FIG. 10 shows that for theraised-triangle depressed cladding fiber when V˜5.31 the group delaycurves for the first four LP modes simultaneously cross resulting in lowDGD across all four low loss modes. However β for the LP₀₂ and LP₂₁modes are small which may result in elevated sensitivity to macrobendsand strong coupling to leaky modes. Since the group delay curves do notsimultaneously cross for larger values of V, this design does notprovide low DGD for a more than four LP modes.

EXAMPLE 1

FIG. 11 shows a raised-triangle profile based on actual VAD core shape.The fiber has 116 ps/km DGD at 1550nm for the LP₀₁, LP₁₁, LP₀₂, and LP₂₁modes.

The following chart gives calculated properties of the raised-triangle,depressed-clad profile shown in FIG. 11:

dgdn wave gd01 gd11 gd02 gd03 gd12 gd21 (ps/m) N01 N11 N02 N03 N12 N211.30 4890.3 4890.0 4891.7 4886.7 4890.7 5.044 1.45114 1.44990 1.44877 01.44737 1.44871 1.31 4890.3 4890.0 4891.7 4886.1 4890.7 5.556 1.451021.44977 1.44863 0 1.44722 1.44857 1.32 4890.3 4890.0 4891.7 4885.54890.7 6.150 1.45090 1.44964 1.44848 0 1.44708 1.44843 1.33 4890.44890.1 4891.7 4884.7 4890.7 6.980 1.45078 1.44951 1.44834 0 1.446941.44828 1.34 4890.4 4890.1 4891.7 4890.8 1.594 1.45065 1.44937 1.44819 00 1.44814 1.35 4890.4 4890.2 4891.7 4890.8 1.554 1.45053 1.44924 1.448050 0 1.44800 1.36 4890.5 4890.2 4891.8 4890.8 1.512 1.45041 1.449111.44791 0 0 1.44786 1.37 4890.6 4890.3 4891.8 4890.9 1.466 1.450281.44898 1.44776 0 0 1.44771 1.38 4890.6 4890.4 4891.8 4891.0 1.4181.45016 1.44885 1.44762 0 0 1.44757 1.39 4890.7 4890.5 4891.8 4891.01.366 1.45004 1.44871 1.44747 0 0 1.44743 1.40 4890.8 4890.6 4891.94892.1 1.311 1.44991 1.44858 1.44733 0 0 1.44728 1.41 4890.9 4890.74891.9 4891.2 1.253 1.44979 1.44845 1.44718 0 0 1.44714 1.42 4891.04890.8 4892.0 4891.3 1.193 1.44967 1.44831 1.44704 0 0 1.44700 1.434891.1 4890.9 4892.0 4891.4 1.128 1.44954 1.44818 1.44689 0 0 1.446851.44 4891.2 4891.0 4892.1 4891.5 1.060 1.44942 1.44805 1.44675 0 01.44671 1.45 4891.3 4891.2 4892.2 4891.6 0.989 1.44929 1.44791 1.44660 00 1.44657 1.46 4891.5 4891.3 4892.2 4891.7 0.914 1.44917 1.44778 1.446460 0 1.44642 1.47 4891.6 4891.5 4892.3 4891.8 0.835 1.44904 1.447641.44631 0 0 1.44628 1.48 4891.8 4891.6 4892.4 4891.9 0.754 1.448921.44751 1.44617 0 0 1.44613 1.49 4891.9 4891.8 4892.5 4892.0 0.6671.44879 1.44737 1.44602 0 0 1.44599 1.50 4892.1 4892.0 4892.5 4892.20.578 1.44866 1.44724 1.44588 0 0 1.44584 1.51 4892.2 4892.1 4892.64892.3 0.484 1.44854 1.44710 1.44573 0 0 1.44570 1.52 4892.4 4892.34892.7 4892.4 0.386 1.44841 1.44697 1.44558 0 0 1.44555 1.53 4892.64892.5 4892.8 4892.6 0.284 1.44828 1.44683 1.44544 0 0 1.44540 1.544892.8 4892.7 4892.9 4892.7 0.178 1.44816 1.44669 1.44529 0 0 1.445261.55 4893.0 4892.9 4893.0 4892.9 0.116 1.44803 1.44656 1.44515 0 01.44511 1.56 4893.2 4893.1 4893.1 4893.0 0.164 1.44790 1.44642 1.44500 00 1.44497 1.57 4893.4 4893.3 4893.2 4893.2 0.244 1.44777 1.44628 1.444850 0 1.44482 1.58 4893.6 4893.5 4893.2 4893.3 0.365 1.44764 1.446141.44470 0 0 1.44467 1.59 4893.8 4893.8 4893.3 4893.5 0.492 1.447511.44600 1.44456 0 0 1.44452 1.60 4894.1 4894.0 4893.4 4893.7 0.6241.44738 1.44586 1.44441 0 0 1.44437 1.61 4894.3 4894.2 4893.5 4893.80.762 1.44725 1.44572 1.44426 0 0 1.44423 1.62 4894.5 4894.5 4893.64894.0 0.905 1.44712 1.44558 1.44411 0 0 1.44408 1.63 4894.8 4894.74893.7 4894.2 1.054 1.44699 1.44544 1.44397 0 0 1.44393 1.64 4895.04895.0 4893.8 4894.4 1.210 1.44686 1.44530 1.44382 0 0 1.44378 1.654895.3 4895.2 4893.9 4894.6 1.372 1.44672 1.44516 1.44367 0 0 1.443631.66 4895.6 4895.5 4894.0 4894.7 1.541 1.44659 1.44502 1.44352 0 01.44348 1.67 4895.8 4895.8 4894.1 4894.9 1.716 1.44646 1.44488 1.44337 00 1.44333 1.68 4896.1 4896.0 4894.2 4895.1 1.899 1.44632 1.44473 1.443220 0 1.44318 1.69 4896.4 4896.3 4894.3 4895.3 2.089 1.44619 1.444591.44307 0 0 1.44303 1.70 4896.7 4896.6 4894.4 4895.5 2.288 1.446061.44444 1.44292 0 0 1.44287 1550 nm properties LP₀₁ LP₁₁ LP₀₂ LP₂₁chromatic dispersion 19.99 20.43 9.15 15.30 (ps/nm · km) effective areaA_(eff) (μm²) 177.1 180.6 353.0 239.5 effective index difference 0.003620.00215 0.00074 0.00071

Since variations on the step-index design such as the raised triangle,depressed-cladding profile can provide only a narrow design space forlow DGD when no more than four modes are allowed to propagate, analternative is to consider Graded Index (GRIN) fiber designs. The indexprofile considered here consists of a graded-index core region and adepressed cladding region (i.e. a “trench”). There could be a number ofadditional design features between the graded core and the trench, suchas a shelf region between the core and the trench or an index stepbetween the core and the trench. The purpose of these features to theindex profile outside the core region is to provide additionalflexibilities to manipulate the spacing of the modal propagationconstants so that the desired combination of transmission properties canbe obtained.

The simplest way to characterize the graded-core region is shown inEq. 1. The alpha parameter α can be chosen between 1 and ∞, whereas α=2corresponds to an inverted parabola. For two-mode design, low DGDbetween LP₀₁ and LP₁₁ modes can be obtained with any α between 1 and ∞combining proper values of other profile parameters such as n_(core),r_(core), trench depth and position. However, for FMF design beyond twoLP modes, α is preferentially chosen close to an inverted parabola shapeto achieve low DGD among all LP modes. The preferred range is2.0+/−0.03. The trench feature has three functions. As shown in FIG. 6,Δβ_(ij) should be as large as possible. A trench structure allowsβ_(min) (LP₁₁ mode in two mode) to be closer to the cladding index whilemaintaining low loss and push down β_(lossy) below the cladding index tobecome a leaky mode. The trench also promotes reduced bending loss anddifferential modal attenuation (DMA) of both LP₀₁ and LP₁₁ modes. Thetrench on the periphery of the raised index core also forms an indexstructure to manipulate DGD, especially of the high order mode(s).

The inventive fiber profiles have a maximum Δ of 0.8%, which will givelower attenuation loss, important for system performance.

EXAMPLE 2

FIG. 12 shows a refractive index profile for Example 2. The fiber has aGRIN core a shelf and a trench. This profile describes a generic classof optical fibers that are particularly effective for multiple modemultiplexing. Profile parameters for the fiber design, describing thecore, shelf, trench, are:

index region start delta end delta alpha width (μm) 1 0.00800 0.001131.972 10.00 2 0.00000 0.00000 0.000 1.92 3 −0.00410 −0.00410 0.000 3.284 0.00000 0.00000 0.000 50.30

1550 nm properties LP₀₁ LP₁₁ LP₀₂ LP₂₁ chromatic dispersion 18.9 19.319.1 19.4 (ps/nm·km) effective area A_(eff) (μm²) 93.2 93.5 186.9 124.6effective index difference 0.0081 0.0053 0.0025 0.0025

FIG. 13 shows the differential group delays for the optical fiber ofExample 2.

As mentioned, the refractive index profile of FIG. 2 is representativeof a family of optical fiber designs that were developed specifically tosupport multiple modes for mode division multiplexing. Many of thesedesigns feature an alpha core, typically having an alpha value between1.5 and 2.5, preferably 1.8 to 2.2, with a truncated edge. A truncatededge is defined as a portion in the refractive index curve that dropsfrom a positive delta index value to zero within two μm or less. Atrench may be similarly defined as having a portion of a refractiveindex curve that abuts or is adjacent to the core, and drops from adelta value at or near zero to a substantial negative value within 2 μmor less. So in general the preferred designs in this category have atruncated alpha core, a ledge, a trench, and an undoped cladding.

It has been found that in some optical fiber designs it may not benecessary to truncate the core. Also it has been found that some designsthat omit the trench may also be effective.

Design parameters for radius width that have been found to be effectiveare:

-   -   Truncated (or standard) core radius: 5 to 20 μm    -   Ledge: 1 to 5 μm    -   Trench: 1 to 10 μm

FIG. 14 gives normalized propagation constant versus V-number for twofamilies of fiber profiles. Solid curves are the simple parabolic (α=2)GRIN profile, while the dashed curves are for a GRIN FMF (dashed lines).The solid curves in FIG. 14 correspond to the α=2.00 GRIN fiber of FIG.4. The dashed curves correspond to a GRIN design with an alpha profilecore and a trench structure which have been optimized for transmissionin the C-band. The modal structure of this new design remains similar tothat of FIG. 4, with some improvement in the normalized propagationconstant. For the same core index, the modes have improved macrobendingcharacteristics, leading to improved DMA. Four LP modes, specificallyLP₀₁, LP₁₁, LP₀₂, and LP₂₁, are well guided at V≈7.5, and the cutoffwavelength of the next higher-order mode LP₁₂ is below 1550 nm. TheV-number is chosen to achieve the large effective index differencebetween the lowest two guided LP modes and the cladding. The largespacing of the normalized propagation constant between different guidedLP modes supports a large Δβ_(lossy), keeping DMA low. FIG. 13 confirmsthe low DGD over a wide bandwidth.

FIG. 15 shows group delays of LP₀₁, LP₁₁, LP₀₂, LP₂₁, LP₀₃ and LP₁₂modes for a parabolic GRIN-FMF (lower curves) and a GRIN-FMF with acladding structure, optimized for operation in the C-band (uppercurves). The vertical line near V-number equal 7.5 corresponds tofour-mode operation.

EXAMPLE 3

A further embodiment of the invention is represented by the refractiveindex profile of FIG. 16. Relevant design parameters are given in thefollowing tables.

index region start delta end delta alpha width (μm) 1 0.00800 0.001071.972 10.00 2 0.00000 0.00000 0.000 1.92 3 −0.00410 −0.00410 0.000 3.284 0.00000 0.00000 0.000 47.30

1550 nm properties LP₀₁ LP₁₁ LP₀₂ LP₂₁ chromatic dispersion 18.5 18.918.6 19.0 (ps/nm·km) effective area A_(eff) (μm²) 92.5 92.8 186.2 123.9effective index difference 0.0081 0.0053 0.0025 0.0025

FIG. 17 gives differential group delay data for the Example 3embodiment.

EXAMPLE 4

A further embodiment of the invention is represented by the refractiveindex profile of FIG. 18. Relevant design parameters are given in thefollowing table.

1550 nm properties LP₀₁ LP₁₁ LP₀₁ LP₂₁ chromatic dispersion 18.9 19.319.2 19.5 (ps/nm·km) effective area A_(eff) (μm²) 86.4 86.8 174.3 115.9effective index difference 0.0079 0.0049 0.0018 0.0018

FIG. 19 gives differential group delay data for the Example 4embodiment.

EXAMPLE 5

A further embodiment of the invention is represented by the refractiveindex profile of FIG. 20. Relevant design parameters and the resulting1550 nm properties are given in the following tables:

index region start delta end delta alpha width (μm) 1 0.0080 0.001091.972 10.00 2 0.0000 0.00000 0.000 1.39 3 −0.0025 −0.00250 0.000 4.00 40.0000 0.00000 0.000 47.11

1550 nm properties LP₀₁ LP₁₁ LP₀₂ LP₂₁ chromatic dispersion 18.9 19.318.1 18.9 (ps/nm·km) effective area A_(eff) (μm²) 92.9 93.2 186.7 124.3effective index difference 0.0081 0.0053 0.0025 0.0025

FIG. 21 gives differential group delay data for the Example 5embodiment.

EXAMPLE 6

A further embodiment of the invention is represented by the refractiveindex profile of FIG. 22. This design is a six-mode design. Relevantdesign parameters are given in the following table.

index region start index end index alpha width 1 0.0127 0.0000 1.9711.30 2 0.0000 0.0000 0.00 0.92 3 −0.0060 −0.0060 0.00 5.00 4 0.00000.0000 0.00 45.28

FIG. 23 gives differential group delay between first five higher orderLP modes and LP₀₁ mode for the design in table above.

The following table shows effective area of LP₀₁, LP₁₁, LP₀₂, LP₂₁, LP₁₂and LP₃₁ modes versus wavelength of the six-mode design in FIG. 22.

wavelength effective area (μm²) (um) LP01 LP11 LP02 LP21 LP12 LP31 1.50090 91 182 121 291 146 1.505 91 91 183 122 292 146 1.510 91 91 184 122293 147 1.515 91 92 184 123 294 147 1.520 92 92 185 123 295 148 1.525 9292 185 123 296 148 1.530 92 93 186 124 297 149 1.535 92 93 187 124 298149 1.540 93 93 187 125 299 150 1.545 93 94 188 125 300 150 1.550 93 94189 125 301 151 1.555 94 94 189 126 302 151 1.560 94 94 190 126 304 1521.565 94 95 190 127 305 152 1.570 95 95 191 127 306 153 1.575 95 95 192128 307 153 1.580 95 96 192 128 308 154 1.585 96 96 193 128 309 1541.590 96 96 193 129 310 155 1.595 96 97 194 129 311 155 1.600 96 97 195130 312 156 1.605 97 97 195 130 314 156 1.610 97 98 196 130 315 1571.615 97 98 197 131 316 157 1.620 98 98 197 131 317 158 1.625 98 98 198132 319 158

Large effective area in optical fibers can reduce nonlinear effects,both intra-modal and inter-modal. In addition, some differential groupdelay can reduce inter-modal nonlinear effect. However pairs of inversedifferential group delay and differential group delay slope with similareffective areas would be desirable to reduce the total span group delayfor simplicity and low cost receiver MIMO design.

In an effort to design very large effective area fibers that supportmultiple independent modes for mode division multiplexing weinvestigated a few-mode fiber design space resulting in an effectivearea large than 160 μm² for LP₀₁ mode, having differential group delaynear zero, tunable for inverse differential group delay and differentialgroup delay slope pairs. We refer to these as ultra large effective areafew-mode fiber designs (ULA-FMF)

EXAMPLE 7 ULA-FMF Design 1

A refractive index profile for ULA-FMF Example 7 is shown in FIG. 24.Numbers for the refractive index profile are:

Region Start Index End Index Alpha Width 1 0.0035 0.0013 1.63 15.0 2−0.0010 −0.0010 0.00 0.6 3 0.0000 0.0000 0.00 46.9

In this ULA-FMF design the delta is kept small for low attenuation.

Differential group delay between LP₁₁ mode and LP₀₁ mode for Example 7is given in FIG. 25.

To illustrate the large effective area of this design the effective areais shown for a range of wavelengths in the following table:

wavelength LP₀₁ Aeff LP₁₁ Aeff (μm) (μm²) (μm²) 1.50 275 276 1.51 277278 1.52 279 279 1.53 281 281 1.54 283 283 1.55 284 285 1.56 286 2871.57 288 289 1.58 290 291 1.59 292 293 1.60 294 294 1.61 296 296 1.62298 298 1.63 299 300

In Example 7 the width of the trench is less than 1 μm. In general,designs with trench widths less than 2 μm for mode division multiplexingare effective and unusual.

EXAMPLE 8 ULA-FMF Design 2

FIG. 26 shows a refractive index profile for a second example of aULA-FMF fiber design.

The design parameters for this example are:

Region Start Index End Index Alpha Width 1 0.0045 0.0000 2.150 15.0 20.0000 0.0000 0.000 47.5

FIG. 27 shows differential group delay between LP₁₁ mode and LP₀₁ modefor the design of Example 8.

The effective area of this design is shown for a range of wavelengths inthe following table:

wavelength LP₀₁ Aeff LP₁₁ Aeff (μm) (μm² ) (μm²) 1.50 209 207 1.51 210208 1.52 211 210 1.53 213 211 1.54 214 213 1.55 216 215 1.56 217 2161.57 218 218 1.58 220 219 1.59 221 221 1.60 222 222 1.61 224 224 1.62225 226

EXAMPLE 9 ULA-FMF Design 3

FIG. 28 shows a refractive index profile for a third example of aULA-FMF fiber design.

The design parameters for this example are:

Region Start Index End Index Alpha Width 1 0.0040 0.0000 1.625 14.0 20.0000 0.0000 0 0.9 3 −0.0060 −0.0060 0 5.0 4 0.0000 0.0000 0 46.9

FIG. 29 shows differential group delay between LP₁₁ mode and LP₀₁ modefor the design of Example 9.

The ULA-FMF design of Example 9 has two guided modes (LP₀₁ and LP₁₁).LP₀₂ mode is cut off at 1.421 μm. The effective area LP₀₁ mode is 196.35μm² at 1550 nm, close to that of ULA-FMF design 2 (215.5 μm² at 1550nm). The DGD slope has opposite signs. Thus ULA-FMF design 2 and 3 couldbe used in pair to reduce total DGD across a total transmission span.The DGD values can be further increased and DGD value/DGD slope can befurther optimized to minimize inter-modal nonlinear effect and reduceaccumulated net total span DGD.

In addition to the application of spatial mode division multiplexingusing both LP₀₁ mode and LP₁₁ mode, single mode launch into LP₀₁ mode isanother potential application. There will be some cross talk due todistributed coupling in long distance transmission, however the smallnet DGD can help to reduce the spread of the distributed coupling intime domain. Few-mode fibers as contemplated for the invention accordingto the current state of the technology generally support from 2 to 10modes. Support in this context means that each of 2 to 10 modes areeffectively transmission channels that are capable of transmittingoptical signals independently without fatal crosstalk. Fatal crosstalkmeans that the signal is degraded beyond intelligence. It is noted thatfor the above Examples 1-9, the effective area of the LP₁₁ and LP₂₁modes should have a scaling factor of 4/3. However, it will beappreciated that this scaling factor does not affect the aspects of theinvention described and claimed herein.

Few-Mode Fibers Supporting 9 or More LP Modes

In the present section, there are provided a number of additionaldetails relating to the design and fabrication of few-mode fibers. Thereare also provided a number of examples of few-mode fibers according toaspects of the invention described above that are capable of supportingnine or more LP modes (i.e., 15 or more spatial modes).

In the above description, it was noted that certain higher-order LPmodes comprise a number of component polarizations, each of which iscapable of carrying a separate spatially-multiplexed signal. Thus,generally speaking, the number of spatial modes supported by a FMF isequal to, or greater than, the number of LP modes supported by the FMF.As used hereinbelow, unless otherwise required by context, the term“mode” by itself refers to an “LP mode” rather than a “spatial mode.”

As discussed above, space-division multiplexing (SDM) transmissionsystems place a number of demands on the transmission fiber. In the caseof a few-mode fiber (FMF), the differential mode loss should be kept toa minimum, the bend loss should be kept to a minimum for the givenapplication, and nonlinearities should be kept to a minimum (i.e., byconfiguring the fiber to have a suitable effective area and dispersion).Furthermore, to reduce the complexity of digital signal processing, itis desirable that the differential group delay (DGD) be kept as low aspossible. Also, in practical transmission systems, splices typicallyoccur every few kilometers. As such, a reasonable demand is that thefiber splices to itself with low loss and low mode coupling.

In the following discussion, there is first described a number of FMFexamples supporting 9 or more LP modes. There is then provided adiscussion of the sensitivity of differential group delay (DGD) withrespect to a number of design parameters.

EXAMPLE 10 9 LP Mode Fiber

In the present section, there is described in detail an FMF designmeeting the above criteria that supports 9 LP modes, allowingmultiplexing over 15 spatial modes.

The maximal difference in DGD for the supported modes was found to beless than 0.8 ps/m, measured with both S² and time-of-flightmeasurement. Strong coupling was observed within the mode groups in thefiber. The attenuation was found to be low for all guided modes, and wasfound to vary between 0.20 and 0.22 dB/km. The design of the describedfiber presented a number of challenges, including very precise controlof the core index profile, to achieve an acceptable amount ofdifferential group delay (DGD). The developed tools described aboveallow for easy optimization of alpha profiles within all MCVD products,decreasing run-in time of new products.

FIGS. 30 and 31 show, respectively, cross section and isometric views ofan exemplary FMF 300 according to the present aspect of the invention,comprising four abutting concentric regions: a graded-index core 301(Region 1), i.e., an “alpha core,” having radius a; an undoped ledgeregion 302 (Region 2); a down-doped trench region 303 (Region 3); and anundoped outer cladding region 304 (Region 4).

FIG. 32 shows a table 320 setting forth the following parameters forRegions 1-4 of the FMF: start index, end index, alpha (α), and width(μm). Generally speaking, for a 9 LP-mode fiber according to the presentinvention, a suitable width for the graded-index core is approximately13 μm or greater.

FIG. 33 shows a refractive index profile 330 for FMF, in which therespective refractive index 331-334 for each of Regions 1-4 is expressedas the index difference relative to the undoped outer cladding region.(By definition, the outer cladding region has an index difference of0.0.)

The graded-index core 301 is designed to minimize differential groupdelay (DGD). The volume of trench 303 is adjusted to minimize bend loss,to ensure that differential mode loss is kept low, and to shift thecutoff wavelength such that exactly 9 LP modes are well-guided: thefundamental LP₀₁ mode and higher-order LP₀₂, LP₀₃, LP₁₁, LP₁₂, LP₂₁,LP₂₂, LP₃₁, and LP₄₁ modes, divided into five mode groups: Group 1(LP₀₁); Group 2 (LP₁₁); Group 3 (LP₀₂, LP₂₁) Group 4 (LP₁₂, LP₃₁); andGroup 5 (LP₀₃, LP₂₂, LP₄₁).

FIG. 34 shows a graph 340 illustrating, for each of the higher-ordermodes, the mode's predicted index difference relative to an undopedsilica cladding, and the mode's predicted DGD. The highest-order modegroup (Group 5: LP₀₃, LP₂₂, LP₄₁) has a maximum effective indexdifference of 2.0·10⁻³ relative to the silica cladding. The fiber ispredicted to have excellent bend performance.

It is noted that the spacing of the mode groups in FIG. 34 correspondsto the schematic diagram of fiber mode propagation constants shown inFIG. 1, discussed above. In particular, the cladding Δn, indicated byvertical broken line 341, corresponds to the horizontal broken line inFIG. 1 labeled “cladding index.” The respective positions of the modegroups along the x-axis, indicated by solid lines 342-345, correspond tothe horizontal solid lines in FIG. 1 labeled “bound modes i,j.”

FIG. 35 shows a table illustrating the theoretical properties of thefiber. Small perturbations to the core profile are predicted to resultin fairly large changes in DGD, and may also yield fiber with negativeDGD, opening up the possibility for all-fiber DGD compensation. Thesensitivity of DGD with respect to a number of design parameters isillustrated in FIGS. 60A-60F, discussed below.

According to a further aspect of the invention, FMF 30 is fabricatedusing a modified chemical vapor deposition (MCVD) technique. A silicatube is loaded into a lathe and is rotated around its longitudinal axisas an aerosol or other vapor-phase deposition technique is used todeposit onto the silica tube's interior wall a series of layers of achemical soot containing silica and one or more selected dopants. Thesilica tube is subsequently sintered and collapsed to form a solidcylindrical preform that is then loaded into a draw tower and drawn intofiber. The concentration of dopant in each deposited layer is controlledby adjusting its flow rate.

In fabricating a prototype of the above-described fiber, an MCVDtechnique was used to construct a preform. The graded-index core 301 wasformed from silica doped with varying amounts of germania in order toobtain the desired profile. The shoulder 302 was formed from undopedsilica. The trench 303 was formed from fluorine-doped silica. Thepreform was then overcladded with a silica jacketing tube and drawn intostandard 125 μm fiber under normal production draw conditions. The outercladding region 304 was formed from the silica substrate tube and thesilica jacketing tube.

FIG. 36 shows a graph 360 illustrating the target refractive indexprofile (curve 361) shown in FIG. 32, and the actual refractive indexprofile (curve 362) achieved in a first attempt to fabricate a preformusing the described MCVD technique.

FIG. 37 shows a graph 370 illustrating the target refractive indexprofile (curve 371) of the graded-index core region, and the measuredrefractive index profile (curve 372). As discussed above, in an MCVDtechnique, the preform is built in successively applied layerscontaining silica and a selected amount of dopant (or no dopant). Theboundary between each layer is marked using a series of broken lines373. The average refractive index for each layer is marked as a seriesof dots 374.

The actual refractive index is reasonably close to the target indexprofile, but in order to obtain good DGD performance, the fabricationtechnique needs to be adjusted. According to a further aspect of theinvention, the index profile of the fabricated fiber was adjusted byusing a second-order polynomial fit of the input germania flow rate tothe average index within each layer.

FIG. 38 shows a graph 380 illustrating the relationship between thepreform index for the graded-index core and the germania flow rate(expressed in arbitrary units). The measured germania flow input data isshown as a series of dots 381. The second-order polynomial fit of theflow input data is shown as a curve 382. The fit was used to approximatenew germania flows based on the differences between target and measuredprofiles. Generally speaking, deviations from the fit within a givenlayer are indicative of temperature fluctuations.

The fiber was characterized by spatially-resolved andspectrally-resolved mode imaging (S² imaging), in order to assessdifferential group delays and mode coupling within the guided modes.FIG. 39 shows a graph 390 of the output of an S² measurement on a20-meter sample of the 9 LP-mode fiber. Both centered-launch (trace 391)and offset-launch (trace 392) were used in order to excite higher-ordermodes. The S² measurements were in reasonable agreement with thecalculated mode delay of the fiber. As expected, significant modecoupling was observed between the modes with similar group indices.

A 13.2 km spool of fiber was then characterized by time-of-flight (ToF)measurements, in order to characterize the multimode fiber impulseresponse and thus the DGD of the fiber. FIG. 40 shows a graph 400 of theresults. The modal content of the individual peaks was resolved byindividually launching each individual mode group, using phase plates ina free space-setup. A CCD camera was used to capture images of the modeintensity, leading to identification of each mode group. In a similarfashion, an optical time-domain reflectometer was used to measure themode dependent loss of each mode group.

The agreement between the S² and time-of-flight measurements was fairlygood, particularly noting that the S² measurement is only on a 20-metersample. Somewhat surprisingly, the LP₃₁/LP₁₂ and LP₄₁/LP₂₂/LP₀₃ groupsseem to couple, despite a fairly large difference in effective index.Simulations from the index profile indicate that this is coincidental.It appears that the apparent coupling of these mode groups havingrespective DGDs that are too similar to each other to distinguisheasily. There is some room for improvement compared to the theoreticaldesign limit.

FIG. 41 shows a table 410 setting forth the measured modal loss for thefour-mode groups in the 9 LP-mode fiber, found by launching each modegroup separately and performing optical time domain reflectometry(OTDR).

Additional exemplary FMFs are described below, including three examplesof a 9 LP mode fiber, a 12 LP mode fiber, a 16 LP mode fiber, and a 20LP mode fiber.

Each of the following exemplary FMF designs comprises four regions: agraded-index core (Region 1); an undoped shoulder region (Region 2); atrench region (Region 3); and an undoped outer cladding region (Region4). The above-described techniques for fabricating FMF 300 are equallyapplicable to the FMFs described below.

EXAMPLE 11-1 9 LP Modes

FIGS. 42-45 set forth data describing another example of a 9 LP mode FMFaccording to the invention.

FIG. 42 shows a table 420 setting forth the following parameters forRegions 1-4 of the FMF: start index, end index, alpha (α), and width(μm). As mentioned above, generally speaking, for a 9 LP-mode fiberaccording to the present invention, a suitable width for thegraded-index core is approximately 13 μm or greater.

FIG. 43 shows the FMF's refractive index profile 430.

FIG. 44 shows a plot 440 of the respective effective index and DGD forthe higher-order modes supported by the FMF.

FIG. 45 shows a table 450 setting forth the respective effective areafor each of the supported modes at wavelengths ranging from 1.485 μm to1.630 μm (in increments of 0.005 μm).

EXAMPLE 11-2 9 LP Modes

FIGS. 46-49 set forth data describing another example of a 9 LP mode FMFaccording to the invention.

FIG. 46 shows a table 460 setting forth the following parameters forRegions 1-4 of the FMF: start index, end index, alpha (α), and width(μm). As mentioned above, generally speaking, for a 9 LP-mode fiberaccording to the present invention, a suitable width for thegraded-index core is approximately 13 μm or greater.

FIG. 47 shows the FMF's refractive index profile 470.

FIG. 48 shows a plot 480 of the respective effective index and DGD forthe higher-order modes supported by the FMF.

FIG. 49 shows a table 490 setting forth the respective effective areafor each of the supported modes at wavelengths ranging from 1.395 to1.690 μm (in increments of 0.005 μm).

EXAMPLE 11-3 9 LP Modes

FIGS. 50-53 set forth data describing a further example of a 9 LP modeFMF according to the invention.

FIG. 50 shows a table 500 setting forth the following parameters forRegions 1-4 of the FMF: start index, end index, alpha (α), and width(μm). As mentioned above, generally speaking, for a 9 LP-mode fiberaccording to the present invention, a suitable width for thegraded-index core is approximately 13 μm or greater.

FIG. 51 shows the FMF's refractive index profile 510.

FIG. 52 shows a plot 520 of the respective effective index and DGD forthe higher-order modes supported by the FMF.

FIG. 53 shows a table 530 setting forth the respective effective areafor each of the supported modes at wavelengths ranging from 1.395 μm to1.640 μm (in increments of 0.005 μm.

EXAMPLE 12 12 LP Modes

FIGS. 54 and 55 set forth data describing another example of a 12 LPmode FMF according to the invention.

FIG. 54 shows a table 540 setting forth the following parameters forRegions 1-4 of the FMF: start index, end index, alpha (α), and width(μm). Generally speaking, for a 12 LP-mode fiber according to thepresent invention, a suitable width for the graded-index core isapproximately 16 μm or greater.

FIG. 55 shows a plot 550 of the respective effective index and DGD forthe higher-order modes supported by the FMF.

EXAMPLE 13 16 LP Modes

FIGS. 56 and 57 set forth data describing another example of a 16 LPmode FMF according to the invention.

FIG. 56 shows a table 560 setting forth the following parameters for theRegions 1-4 of the FMF: start index, end index, alpha (α), and width(μm). Generally speaking, for a 12 LP-mode fiber according to thepresent invention, a suitable width for the graded-index core isapproximately 19 μm or greater.

FIG. 57 shows a plot 570 of the respective effective index and DGD forthe higher-order modes supported by the FMF.

EXAMPLE 14 20 LP Modes

FIGS. 58 and 59 set forth data describing another example of a 20 LPmode FMF according to the invention.

FIG. 58 shows a table 580 setting forth the following parameters forRegions 1-4 of the FMF: start index, end index, alpha (α), and width(μm). Generally speaking, for a 20 LP-mode fiber according to thepresent invention, a suitable width for the graded-index core isapproximately 22 μm or greater.

FIG. 59 shows a plot 590 of the respective effective index and DGD forthe higher-order modes supported by the FMF.

Sensitivity of DGD with Respect to Various Design Parameters

Generally speaking, DGD varies as a function of a number of fiber designparameters, including core dopant profile concentration, core diameter,core alpha, core delta, shoulder width, trench delta, and trenchthickness.

The sensitivity of DGD to the above parameters was investigated withrespect to an exemplary two-mode fiber design supporting propagation ofthe LP₀₁ and LP₁₁ modes. The following table sets forth the simulationparameters used in the investigation:

Col. 1 Col. 2 Col. 3 Col. 4 Parameter Design Lowered Raised A. CoreDopant Concentration (FIG. 60A) 1.00 (nominal) 0.98 1.02 B. CoreDiameter (FIG. 60B) 7.47 μm 7.37 μm 7.57 μm C. Core Alpha (FIG. 60C)2.014 2.004 2.024 D. Core Delta (FIG. 60D) 0.95% 0.90% 1.00% E. ShoulderWidth (FIG. 60E) 0.6544 μm 0.5544 μm 0.7544 μm F. Trench Delta (FIG.60F) −0.41% −0.39% −0.43% G. Trench Thickness (not shown) 5 μm 5 μm (nochange) 5 μm (no change)

Column 1 sets forth the parameters that were investigated; Column 2 setsforth the design parameters for the exemplary two-mode FMF; Column 3sets forth a selected lowered parameter value; and Column 4 sets forth aselected raised parameter value.

FIGS. 60A-60F are a series of figures illustrating the relationshipbetween DGD and wavelength for the LP₀₁ and LP₁₁ modes for each of theabove parameters. In each of these figures, the DGD data for theunmodified fiber is shown as a central plot; the DGD data for thelowered parameter is shown as a plot below the central plot; the DGDdata for the raised parameter is shown as a plot above the central plot.

While the foregoing description includes details which will enable thoseskilled in the art to practice the invention, it should be recognizedthat the description is illustrative in nature and that manymodifications and variations thereof will be apparent to those skilledin the art having the benefit of these teachings. It is accordinglyintended that the invention herein be defined solely by the claimsappended hereto and that the claims be interpreted as broadly aspermitted by the prior art.

1. An optical fiber, comprising: a core and a cladding surrounding thecore, wherein the core and cladding have a refractive index profile thatis structured to support propagation of a plurality of desiredsignal-carrying modes, while suppressing undesired modes, wherein thecore comprises a graded-index core, wherein the cladding comprises aledge between the core and the trench, wherein the cladding furthercomprises a down-doped trench abutting the ledge, and an undopedcladding region abutting the trench, wherein the core, and cladding areconfigured to support propagation of a spatially-multiplexed opticalsignal comprising a plurality of desired modes, while suppressingundesired modes, wherein the core and surrounding cladding areconfigured such that undesired modes have respective effective indicesthat are close to or less than the cladding index so as to result inleaky modes that leak into the outer cladding region, and wherein theindex spacing between the desired mode having the lowest effective indexand the leaky mode with the highest effective index is sufficientlylarge so as to substantially prevent coupling therebetween, wherein theoptical fiber is a few-mode fiber designed for 9-20 modes.
 2. Theoptical fiber of claim 1, wherein the fiber is designed for 9 modes andcomprises a graded-index core region having a radius of 13 μm orgreater.
 3. The optical fiber of claim 1, wherein the fiber is designedfor 12 modes and comprises a graded-index core region having a radius of16 μm or greater.
 4. The optical fiber of claim 1, wherein the fiber isdesigned for 16 modes and comprises a graded-index core region having aradius of 19 μm or greater.
 5. The optical fiber of claim 1, wherein thefiber is designed for 20 modes and comprises a graded-index core regionhaving a radius of 22 μm or greater.